Hilbert transforms along Lipschitz direction fields: A lacunary model

Shaoming Guo; Christoph Thiele

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arXiv · 1603.03317

Hilbert transforms along Lipschitz direction fields: A lacunary model

Shaoming Guo·Christoph Thiele

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Abstract

We prove bounds for the truncated directional Hilbert transform in $L^p(\mathbb{R}^2)$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for $p=2$, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some $p$.

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Shaoming Guo, Christoph Thiele. 2016-03-10. Hilbert transforms along Lipschitz direction fields: A lacunary model. https://arxiv.org/abs/1603.03317

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